A segment connecting the midpoints of two sides of a triangle is parallel to the third side and exactly half its length.
Picture a triangular picture frame hanging on a wall. Stretch a rubber band between the midpoints of two sides. That rubber band runs perfectly parallel to the bottom of the frame, like a miniature shelf—and it spans exactly half the width. No matter how you reshape the triangle, that halfway connection always mirrors the opposite side at half scale.
Showing a random 20 of 50 problems.
Example 1
easy
The midsegment of a triangle has length 14. What is the length of the side parallel to the midsegment?MN is the midsegment parallel to BC. MN = 14; find BC.
Example 2
easy
Triangle XYZ has midsegment PQ where P is the midpoint of XY and Q of XZ. If YZ=18, find PQ.
Example 3
medium
In △PQR, M is the midpoint of PQ and N is the midpoint of QR. If MN=3x−1 and PR=4x+6, find the value of x and the length MN.M is the midpoint of PQ and N is the midpoint of QR. MN = 3x − 1; PR = 4x + 6.
Example 4
easy
A triangle has third side of length 40. The midsegment parallel to that side has length ℓ. Find ℓ and explain why.
Example 5
hard
In △ABC with BC=24, the midsegment MN parallel to BC is extended (along the line through M and N) to a longer chord through the triangle. Why is the extended chord still less than 24 if it stays parallel to BC and inside the triangle?
Example 6
medium
In quadrilateral context: connecting midpoints of all four sides of any quadrilateral forms what shape, and why (Varignon's theorem)?
Example 7
easy
The third side of a triangle is 18. How long is the midsegment parallel to it?
Example 8
challenge
Using coordinates, prove the midsegment theorem for a triangle with vertices A(0,0), B(2b,0), C(2c,2d).
Example 9
medium
The medial triangle has area 7. What is the area of the original triangle?
Example 10
hard
In △ABC, the three midsegments are drawn, dividing the triangle into four smaller triangles. If the area of △ABC is 120 cm², what is the area of each smaller triangle? Justify using the Midsegment Theorem.The three midsegments divide triangle ABC (area 120 cm²) into four congruent triangles. Find the area of each.
Example 11
challenge
Prove that in any quadrilateral, the four midpoints of the sides form a parallelogram whose area is half the area of the original quadrilateral.
Example 12
easy
A triangle's third side has length 24. How long is the midsegment parallel to it?
Example 13
medium
In triangle ABC, the midsegment DE (D on AB, E on AC) is parallel to BC. If ∠ADE=65°, what is ∠ABC?
Example 14
medium
A trapezoid's midsegment (median) connects midpoints of the two legs. If the parallel sides are 8 and 14, find the midsegment length.
Example 15
easy
A midsegment is 9. How long is the side it is parallel to?
Example 16
hard
In △ABC, let G be the centroid. A midsegment MN is drawn parallel to BC. What fraction of the way from A to BC does G lie?
Example 17
medium
In triangle ABC, D and E are midpoints of AB and AC. If DE = 2x+1 and BC = 5x-2, find x.
Example 18
hard
In △ABC, the medial triangle has area 5 cm2. Find the area of each of the three corner triangles formed by the midsegments.
Example 19
challenge
Explain why the medial triangle is congruent to each of the three corner triangles formed by the midsegments.
Example 20
medium
D is the midpoint of AB, E is the midpoint of AC, DE=8 and is parallel to BC. A line parallel to BC is drawn 3/4 of the way from A to BC. Find its length.